Repairing from the Input Language.
This approach synthesizes a new transducer for the inputs on which \(T_{\textit {bad}}\)  is incorrect.
Using properties of transducers, we can compute an automaton describing the exact set of inputs \(P_{\textit {bad}}\subseteq P\)  for which \(T_{\textit {bad}}\)  does not produce an output in \(Q\)  (see pre-image computation in [1]}).
Let \(restrict(T,L)\)  be the transducer that behaves as \(T\)  if the input is in \(L\)  and does not produce an output otherwise (transducers are closed under restriction [1]}).
If we synthesize a transducer \(T_1\)  with type \(\lbrace P_{\textit {bad}}\rbrace T_1\lbrace Q\rbrace \) , then the transducer \(restrict(T_1,P_{\textit {bad}})\cup restrict(T_{\textit {bad}},P\setminus P_{\textit {bad}})\)  satisfies the desired input-output type (transducers are closed under union).
